SIMULATION STUDY 1: PROPERTIES OF THE EMFG TEST
For Simulation Study 1, each of 1,000 simulated data sets contain 200 three-generational pedigrees. Only individuals 7 and 8 are affected (with dark shading in ). First, we use the EMFG Test to estimate the MFG effect in simulated samples with no gender-specific MFG effect (, Scenarios A and B). When μ = 1, the type-I error rate of 0.040 is close to the desired level of 0.05. The EMFG test has 0.816 power to detect an MFG effect of 1.7 when there are no gender effects (Scenario B, model μm
). When these same data sets are analyzed under a model that allows for gender differences (μm
), power is reduced to 0.724. In all scenarios, the parameter estimates are close to the true MFG effect. Founder genotype frequencies and their standard errors are the same across models, and are equal to their expected values under HWE (
(1/1) = 0.109, SE = 0.012;
(1/2) = 0.443, SE = 0.018;
(2/2) = 0.448, SE = 0.018). Accurate results are also obtained for data simulated with 1.5≤μ≤2.5 (data not shown).
Pedigree structure used for Simulation Studies 1 and 2. Dark shading represents individuals who are affected in both simulations (Simulation Studies 1 and 2). Hatching represents individuals who are only affected in Simulation Study 2.
Properties of the EMFG test simulation study 1—RHD incompatibility
Several studies found that MFG incompatibility effects are confined to a single gender [Insel et al., 2005
; Palmer et al., 2006
]. To mimic this scenario, samples are simulated with males at increased risk of disease but females at baseline risk (μf
= 1.0). Under a correct model constraining the female effect to 1, power to detect the MFG effect is 0.536 (, Scenario C). Power decreases because each family has exactly one affected male and one affected female, so there are now only half the individuals contributing to μm
’s estimation. Under a model that allows gender differences, parameter estimates are close to the true values and power is reduced to 0.45. Genotype frequency estimates and standard errors are the same as those observed in the previous scenarios. Similar results are obtained for data simulated with any value of μm
between 1.5 and 2.5 (data not shown).
To determine type I error, power, coverage, and parameter estimation of the EMFG when offspring allelic effects are present, we simulated data with NIMA MFG incompatibility and offspring allelic effects (, Column 5). A total of 15% of RA patients do not carry the shared epitope encoded by certain HLA-DRB1 alleles [Harney et al., 2003
], so bi-allelic gene frequencies are P
(1) = 0.39 and P
(2) = 0.61 where allele 2 represents the shared epitope. Data are simulated without NIMA (μ = 1) or offspring effects (ρ1
= 1) (, Scenario A), and with μ = 2.5, ρ1
= 2 and ρ2
= 2.5 (Scenario B). In both scenarios, we fit a full model that estimates NIMA and offspring allelic effects. We also fit three reduced models (results not shown), that represent the null models of (a) no offspring allelic or NIMA effects (μ = ρ1
= 1.0), (b) no offspring allelic effects (ρ1
= 1.0), and (c) no NIMA effect (μ = 1.0). Using LR tests, these three models allow us to test for (a) either offspring or NIMA effects, (b) offspring effects in the presence of NIMA effects, and (c) NIMA effects in the presence of offspring effects.
Properties of the EMFG tests simulation study 1—NIMA incompatibility
When simulated under the null hypothesis (μ = ρ1
= 1.0), the parameter estimates (1.001–1.014), coverages (0.941–0.956), and type-I error rates (0.048–0.060) are appropriate. When μ = 2.5, ρ1
= 2.0, and ρ2
= 2.5, parameter estimates are close to their true values (
= 2.57, 1
= 2.058, 2
= 2.575) and coverage is appropriate (0.94–0.95). There is ~80% power to detect all three effects (rejection rate = 0.818 of μ = 1.0, rejection rate = 0.771 of ρ1
= 1.0, and rejection rate = 0.797 of μ = ρ1
= 1.0). Founder genotype frequencies are equal to their expected values under HWE.
SIMULATION STUDY 2: COMPARISON OF THE EXTENDED PEDIGREE AND NUCLEAR FAMILY STUDY DESIGNS
To avoid potential biases in analyzing related and sometimes overlapping nuclear families, the current recommended practice when using the MFG test is to select nuclear families from an extended pedigree so that the offspring in different nuclear families are no more related than second cousins [Palmer et al., 2008
]. To examine the impact on power, we consider three simulation scenarios involving 80 three-generational pedigrees with affected siblings and first cousins (, individuals 5–8). reports parameter estimates, standard errors, coverage, and rejection rates using the EMFG test. The 80 three-generational pedigrees are analyzed as full pedigrees (Scenario A), as 80 selected nuclear families by removing individuals 3, 4, 7, and 8 (Scenario B), and as 240 independent nuclear families (Scenario C). Regardless of which we use, all the three approaches give reasonable parameter estimates (
range: 0.979–0.993), type-I error rates (~0.060), and coverage (range: 0.942–0.965) when μ = 1.0.
Extended pedigrees in their entirety vs. partitioning pedigrees into nuclear families
When μ = 1.7, analyzing pedigrees in their entirety produces the most accurate result (
= 1:691, coverage = 0.953). Analyzing either 80 nuclear families or all 240 nuclear families slightly underestimates the MFG effect (Scenario B:
= 1.598, coverage = 0.948; Scenario C:
= 1.623, coverage = 0.932). As expected, analyzing pedigrees in their entirety is more powerful than analyzing one nuclear family per pedigree, but analyzing all 240 nuclear families recovers the power (Scenario A: rejection rate = 0.825; Scenario B: rejection rate = 0.607; and Scenario C: rejection rate = 0.80). More striking trends are seen when μ = 2.5 where there is plenty of power to detect the MFG effect with all three study designs (power > 0.999). Analyzing pedigrees in their entirety gives the most accurate estimates (
= 2.532, coverage = 0.954). Surprisingly, we see decided underestimation and reduced coverage when analyzing either one nuclear family per pedigree or analyzing all nuclear families (Scenario B:
= 2.231, coverage = 0.900; Scenario C:
= 2.356, coverage = 0.913).
To investigate if the parameter underestimation observed in Scenarios B and C results from the random mating assumption, we analyzed the same samples simulated in Scenario C (μ = 1.7 and 2.5) using the original mating type based nuclear family MFG test (Scenario D). Under both values of μ, the parameters are again underestimated (
= 1.636, coverage = 0.940;
= 2.376, coverage = 0.926). Since we continue to see underestimation of the MFG parameter even when we do not assume random mating, we can rule it out as a contributing factor. When μ = 1.7, the rejection rate is slightly lower than in Scenario C, and this difference proves to be statistically significant according to McNemar’s test (P
-value < 0.001). This suggests that assuming random mating when it holds increases the power of the MFG test.
Instead, the underestimation is the result of the loss of information regarding MFG incompatibility that occurs when analyzing extended pedigrees as nuclear families. For our simulations, in order for individual 8 to be MFG incompatible, her mother (individual 5) must have genotype 1/1, meaning that the individual 5 cannot be MFG incompatible with her mother (individual 2). As a result, we expect individual 5 to carry the 1/1 genotype more often and carry the 1/2 genotype less often than individual 6, whose child (individual 7) can be incompatible with his mother (individual 3) even when the individual 6 is incompatible with his mother (individual 2). We can make our intuition more concrete by calculating the average frequencies for individuals 5 and 6 over all the simulations under Scenario A where μ = 2.5. The frequency of the 1/1 genotype for individual 5 (f1/1 = 0.166) is greater than the frequency for individual 6 (f1/1 = 0.109), and the frequency of the 1/2 genotype for individual 5 (f1/2 = 0.523) is less than the frequency for individual 6 (f1/2 = 0.664). When analyzed as a nuclear family, individual 6’s decreased tendency to be incompatible with individual 2, leads to an underestimation in μ. On the other hand, the extended pedigree’s likelihood correctly accounts for the relationships among the three generations.
There are additional situations where exploiting extended pedigrees has advantages. For example, if a parent of an affected individual is unavailable, we expect an increase in power if we genotype the parents of the missing parent. Using the 80 three-generational pedigrees from , Scenario A, we deleted genotypes in the second generation so that 40 families are missing individuals 3–6, 20 families are missing individuals 5 and 6, 10 families are missing individual 5, and 10 families are missing individual 6. After analyzing the 80 extended pedigrees with missing genotypes (, Scenario E), we conducted two nuclear family analyses. In the first analysis (Scenario F), we select a nuclear family with at least one genotyped affected offspring from each pedigree, resulting in 80 nuclear families, choosing trios over dyads, and dyads over singletons (individuals with two ungenotyped parents). If two nuclear families fit this criterion, one is chosen at random. Our final sample composition is 20 trios, 20 dyads, and 40 singletons. In the second nuclear family analysis (Scenario G), we analyze all nuclear families with at least one genotyped affected offspring, for a total of 180 nuclear families (40 trios, 60 dyads, and 80 singletons).
When μ = 1.0, parameter estimates
(0.972–1.001), type-I error rates (0.047–0.057), and coverage (0.952–0.961) are appropriate if extended pedigrees are analyzed in their entirety, if a single nuclear family per pedigree is analyzed, or if all allowable nuclear families are analyzed. The most dramatic difference from the complete genotype data analyses are the increases in the SEs for μ when the pedigrees with missing data are analyzed as nuclear families (Scenario F: SE = 1.923 vs. Scenario B: SE = 0.277; Scenario G: SE = 0.680 vs. Scenario C: SE = 0.211), whereas the SE for μ is only slightly increased when using an extended pedigree study design (Scenario E: SE = 0.252 vs. Scenario A: SE = 0.219).
When μ = 1.7, the EMFG test still yields reasonable parameter estimates and power in the presence of missing genotypes when extended pedigrees are analyzed in their entirety (
= 1.697, coverage = 0.959, rejection rate = 0.736). There is a slight power loss compared to the situation where all individuals are genotyped (Scenario A, rejection rate = 0.825). When a single nuclear family per pedigree is analyzed, there is a large reduction in power relative to Scenario B, and much larger standard errors (1.585), but μ is only slightly underestimated (
= 1.612, coverage = 0.974, rejection rate = 0.222). When all allowable nuclear families are analyzed (Scenario G), μ is slightly underestimated, and power is larger in comparison to using only one nuclear family per analysis, but is still much smaller than the power seen when analyzing extended pedigrees (
= 1.624, coverage = 0.960, rejection rate = 0.359). The SE of μ is still large (0.504) but not nearly as large as in Scenario F. Thus, analyzing extended pedigrees provides a dramatic increase in power over nuclear families when there are missing data.
When μ = 2.5 and pedigrees are analyzed in their entirety, parameter estimates are accurate, and power is very high (
= 2.536, coverage = 0.957, rejection rate ≥ 0.999). When nuclear families are analyzed, the MFG parameter is underestimated, standard errors are large, and power is reduced (Scenario F:
= 2.31, SE = 0.974, coverage = 0.972, rejection rate = 0.549; Scenario G:
= 2.306, SE = 0.627, coverage = 0.939, rejection rate = 0.853). Again the large standard errors for μ offset the bias so that the coverage is not significantly reduced from the nominal value of 0.95.
SIMULATION STUDY 3: POPULATION STRATIFICATION
Because the EMFG test assumes random mating it is important to investigate violations of this assumption. Violations can occur in different ways. For example, population stratification (PS) may be present. Sinsheimer et al. 
found that the original MFG test, which does not assume random mating, is unaffected by PS, but it is possible that the EMFG test may be sensitive to it.
To understand the impact of PS, we turn to additional simulations. Each scenario involves one of three levels (No PS, Moderate PS, and Large PS), 1,000 samples, and the null hypothesis of μ = 1.0 or the alternative hypothesis of μ = 2.5. The true MFG effect is the same in the population but the baseline disease risk differs between populations. To allow direct comparison to the nuclear family MFG test, we simulate 240 parent-offspring trios. In the No PS example, trios are sampled from one population using an average of RHD frequencies for Finland and Germany (P
(1) = 0.37, P
(2) = 0.63). In the second example (Moderate PS), half the trios have Finnish RHD frequencies [Palmer et al., 2002
(1) = 0.33, P
(2) = 0.67], and half have German RHD frequencies [Wagner et al., 1995
(1) = 0.41, P
(2) = 0.59]. In the third example (Large PS), half the trios have Finnish RHD frequencies, and half have Asian RHD frequencies [Reid and Lomas-Francis, 2004
(1) = 0.10, P
(2) = 0.90]. Results are shown in .
Parameter estimates, coverage, and rejection rates for Simulation Study 3
For the samples with μ = 1.0 and No PS or Moderate PS, coverage (0.954 and 0.953), type-I error rate (0.050 and 0.052), and parameter estimation (0.983 and 0.970) are appropriate. Under large PS, type-I error rate is inflated (0.085), coverage is low (0.929), and μ is underestimated (0.804). For samples with μ = 2.5 and No PS (
= 2.506, coverage = 0.960, power = 0.988) or Moderate PS (
= 2.48, coverage 50.949, power = 0.985), the results are very similar. In the Large PS scenario, MFG is underestimated (2.054), coverage is low (0.900), and power is greatly reduced (rejection rate = 0.636). If we analyze these same samples using the nuclear family MFG test [Kraft et al., 2004
], we obtain an appropriate parameter estimate and coverage (
= 2.492, coverage = 0.970). However, power is reduced (rejection rate = 0.768), reflecting the small proportion of Asians who are affected because of MFG incompatibility.
Because the EMFG test is sensitive to random mating violation, we suggest testing for departures from this assumption in a randomly selected sample of control parent pairs from the same study population. One option is to test for departure from HWE. A better option is to test for random mating violation by comparing observed parental mating type frequencies to expected mating type frequencies assuming random mating (). Let P
represent the probability of mating type j
under the null (random mating holds) and alternative (random mating violated) hypotheses. If we denote the corresponding multinomial mating type counts in the control data by Nj
, then the appropriate test statistic is
This statistic is compared to a χ2 distribution with three degrees of freedom. When we apply the HWE and LR tests to 1,000 samples each comprised of 120 Finnish control parents and 120 Asian control parents (, Scenario C), non-random mating that leads to 20% underestimation of μ (, Scenario D) can be detected with approximately 70% power under the LR test (5) vs. 40% power under a HWE test. The random mating test is more powerful despite having more degrees of freedom and allows us to determine situations where the nuclear family MFG test may be preferable to the EMFG test.